This question orginally comes from the exercise 3.18 of Hartshorne, Algebraic Geometry. If $Y$ is a projective variety in $\mathbb{P}^n$ then $Y$ is projectively normal (w.r.t the embedding) if its homogenoeus coordinate ring $S(Y)$ is integrally closed.

We can easily prove that if $Y$ is projectively normal then $Y$ is normal. However, the converse, as Hartshorne claimed, is not true. He gave as an example the twisted quartic in $\mathbb{P}^3$.

My question are :

How can I prove that the twisted quartic is not normal ? I attempt to show that its homogenoeus coordinate ring is not integrally closed, but I could not find the counter example. Is there other way to show that the twisted quartic is not normal ?

Is projectively normal important ? I searched google for the information about that property, and it gave me alot of information related to algebraic geometry. What is its role in algebraic geometry ?

Could you please show me some other example of normal variety in some projective space that is not projectively normal?